What is the integral of 2x(x^2+5)^{-4} ?

The integral of 2x(x^2+5)^{-4} is -1/(3(x^2+5)^3)+C

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integral of 2x(x^2+5)^{-4}

\int 2 x (x^{2} + 5)^{- 4} d x

- \frac{1}{3 ( x ^{2} + 5 ) ^{3}} + C

Solution steps

\int 2 x (x^{2} + 5)^{- 4} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x (x^{2} + 5)^{- 4} d x

Apply u-substitution

: \int \frac{1}{2 u ^{4}} d u

= 2 \cdot \int \frac{1}{2 u ^{4}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u ^{4}} d u

Apply the Power Rule

: - \frac{1}{3 u ^{3}}

= 2 \cdot \frac{1}{2} (- \frac{1}{3 u ^{3}})

Substitute back

u = x^{2} + 5

= 2 \cdot \frac{1}{2} (- \frac{1}{3 ( x ^{2} + 5 ) ^{3}})

Simplify

2 \cdot \frac{1}{2} (- \frac{1}{3 ( x ^{2} + 5 ) ^{3}}) : - \frac{1}{3 ( x ^{2} + 5 ) ^{3}}

= - \frac{1}{3 ( x ^{2} + 5 ) ^{3}}

Add a constant to the solution

= - \frac{1}{3 ( x ^{2} + 5 ) ^{3}} + C

x^{2} + 6, 12 + 4 x - x^{2}

n = 1 \sum \infty 0. 4^{n}

f (x) = x^{3} - x^{2}

f (x) = 4 x^{2} + 2

n = 1 \sum \infty \frac{1}{n 7 ^{n}}

What is the integral of 2x(x^2+5)^{-4} ?
The integral of 2x(x^2+5)^{-4} is -1/(3(x^2+5)^3)+C